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AbstractLet p1, p2 be two distinct prime integers, let n be a positive integer, n≥3 and let ξn be a primitive root of order n of the unity. In the 3rd section of this paper we obtain a complete characterization for a quaternion algebra H(p1,p2) to be a division algebra over the nth cyclotomic field Q(ξn), when n∈{3,4,6,7,8,9,11,12} and we also obtain a characterization for a quaternion algebra H(p1,p2) to be a division algebra over the nth cyclotomic field Q(ξn), when n∈{5,10}. In the 4th section we obtain a complete characterization for a quaternion algebra HQ(ξn)(p1,p2) to be a division algebra, when n=lk, with l a prime integer, l≡3 (mod 4) and k a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra HQ(ξl)(p1,p2) to be a division algebra, when l is a Fermat prime number. [ABSTRACT FROM AUTHOR]

Let k = Q ( p 1 p 2 q 1 q 2 ) be a real quadratic number field, where p i ≡ - q i ≡ 1 (mod 4) , i = 1 , 2 , are different prime integers and C k , 2 its 2-class group. Let k 2 (1) (resp. k 2 (2) ) be the first (resp. second) Hilbert 2-class field of k . In this article, we investigate the metacyclicity of G = G a l (k 2 (2) / k) and the cyclicity of G ′ = G a l (k 2 (2) / k 2 (1)) , the derived subgroup of G, assuming C k , 2 ≃ G / G ′ ≃ Z / 2 Z × Z / 2 n Z , with n ≥ 2 . As application we study the capitulation of C k , 2 in the quadratic and biquadratic subfields of k 2 (1) . [ABSTRACT FROM AUTHOR]

The periodic points of the algebraic function defined by the equation g (x , y) = x 3 (4 y 2 + 2 y + 1) - y (y 2 - y + 1) = 0 are shown to be expressible in terms of values of Ramanujan's cubic continued fraction c (τ) with arguments in an imaginary quadratic field K in which the prime 3 splits. If w = (a + - d) / 2 lies in an order of conductor f in K and 9 ∣ N K / Q (w) , then one of these periodic points is c(w/3), which is shown to generate the ring class field of conductor 2f over K. [ABSTRACT FROM AUTHOR]

Motivated by the recent work of Park on the analogue of the Ramanujan’s function k (τ) = r (τ) r 2 (2 τ) for the Ramanujan’s cubic continued fraction, where r (τ) is the Rogers–Ramanujan continued fraction, we use the methods of Lee and Park to study the modularity and arithmetic of the function w (τ) = X (τ) X (3 τ) , which may be considered as an analogue of k (τ) for the continued fraction X (τ) of order six introduced by Vasuki, Bhaskar and Sharath. In particular, we show that w (τ) can be written in terms of the normalized generator u (τ) of the field of all modular functions on Γ 0 (18) , and derive modular equations for u (τ) of smaller prime levels. We also express j (d τ) for d ∈ { 1 , 2 , 3 , 6 , 9 , 18 } in terms of u (τ) , where j is the modular j-invariant. [ABSTRACT FROM AUTHOR]

Let p ≡ 1 (mod 8) and q ≡ 7 (mod 8) be two prime numbers. The purpose of this paper is to compute the unit groups of the fields L = Q (2 , p , q) and give their 2-class numbers. [ABSTRACT FROM AUTHOR]

Let K/F be a finite extension of number fields and S be a finite set of primes of F, including all the Archimedean ones. In this paper, using some results of González-Avilés (J Reine Angew Math 613:75–97, 2007), we generalize the notions of the relative Pólya group Po (K / F) (Chabert in J Number Theory 203:360–375, 2019; Maarefparvar and Rajaei in J Number Theory 207:367-384, 2020) and the Ostrowski quotient Ost (K / F) (Shahoseini et al. in Pac J Math 321(2):415–429, 2022) to their S-versions. Using this approach, we obtain generalizations of some well-known results on the S-capitulation map, including an S-version of Hilbert’s Theorem 94. [ABSTRACT FROM AUTHOR]

For an imaginary biquadratic number field K = Q (- q , d) , where q > 3 is a prime congruent to 3 (mod 8) , and d is an odd square-free integer which is not equal to q, let K ∞ be the cyclotomic Z 2 -extension of K . For any integer n ≥ 0 , we denote by K n the nth layer of K ∞ / K . We investigate the rank of the 2-class group of K n , then we draw the list of all number fields K such that the Galois group of the maximal unramified pro-2-extension over their cyclotomic Z 2 -extension is metacyclic pro-2 group. [ABSTRACT FROM AUTHOR]

We apply the methods of Lee and Park to study a certain continued fraction H (τ) of Ramanujan, which is a particular case of his identity written in one of his notebooks. We prove that H (τ) can be expressed in terms of an η -quotient s (τ) , which is a generator for the field of all modular functions on the congruence subgroup Γ 0 (12) . We also show that there is a modular equation for s (τ) and H (τ) of level n for all positive integers n, and provide explicitly the modular equations of level p for primes p ≤ 13 . [ABSTRACT FROM AUTHOR]

A formula is proved for the number of linear factors and irreducible cubic factors over F l of the Hasse invariant H ^ 7 , l (a) of the elliptic curve E 7 (a) in Tate normal form, on which the point (0, 0) has order 7, as a polynomial in the parameter a, in terms of the class number of the imaginary quadratic field K = Q (- l) . Conjectural formulas are stated for the numbers of quadratic and sextic factors of H ^ 7 , l (a) of certain specific forms in terms of the class number of Q (- 7 l) , which are shown to imply a recent conjecture of Nakaya on the number of linear factors over F l of the supersingular polynomial s s l (7 ∗) (X) corresponding to the Fricke group Γ 0 ∗ (7) . [ABSTRACT FROM AUTHOR]

In this paper, we present an alternative method to compute all imaginary octic non-CM fields with class number one, listed in Yamamura’s paper (Acta Arith 86:133–47, 1998). We consider such octic fields as an abelian extension over an appropriate quadratic subfield, whereas Yamaura’s method uses a known structure for quartic fields. In fact, we demonstrate that all such imaginary non-CM octic fields with class number one are subfields of the ray class fields of some imaginary quadratic fields with an appropriate conductor. Using the Pari software, as described in Sect. 3, we determine all imaginary octic non-CM fields with class number one. (The codes are included as an appendix.) [ABSTRACT FROM AUTHOR]

Let n be an integer ≥ 1 and p 1 ,... , p n distinct odd prime integers. In this article, we give the list of all imaginary multiquadratic number fields K n = Q (i , 2 , p 1 , ... , p n) that have a cyclic 2-class group. [ABSTRACT FROM AUTHOR]

It is well known from the results of Ferrero and Kida [2,7] that the Z 2 -torsion part of the unramified abelian Iwasawa module X ∞ of any imaginary quadratic number field is trivial or cyclic of order 2. We will determine an infinite family of CM number fields, in which the Z 2 -torsion of the Iwasawa module X ∞ is of arbitrary large rank, giving also the exact value of the rank of X ∞ . [ABSTRACT FROM AUTHOR]

For a metabelian 2-group G such that G / G ′ ≃ (2 , 2 n) , n ≥ 2 , we are interested in giving necessary and sufficient conditions to have G abelian, modular, metacyclic non modular or nonmetacyclic. The abelianizations of the subgroups of index 2 and 4, and the kernel of the transfer map in the metacyclic case are given too. As an application of these results, we investigate the capitulation of 2-classes of fields whose 2-class groups are of type (2 , 2 n) , n ≥ 2 . Examples are given by considering some real quadratic fields. [ABSTRACT FROM AUTHOR]

Let KK be an imaginary quadratic field different from Q(−1){\mathbb{Q}}\left(\sqrt{-1}) and Q(−3){\mathbb{Q}}\left(\sqrt{-3}). For a nontrivial integral ideal m{\mathfrak{m}} of KK, let Km{K}_{{\mathfrak{m}}} be the ray class field modulo m{\mathfrak{m}}. By using some inequalities on special values of modular functions, we show that a single xx-coordinate of a certain elliptic curve generates Km{K}_{{\mathfrak{m}}} over KK.

When singular moduli of Hauptmodules generate ring class fields (resp. ray class fields) of imaginary quadratic fields, using the theory of Shimura reciprocity law, we determine a necessary and sufficient condition for singular moduli of rrth roots of the Hauptmodules to generate the same ring class fields (resp. ray class fields) as do the singular moduli of the Hauptmodules.

An abelian threefold A / Q of prime conductor N is favorable if its 2-division field F is an S 7 -extension over Q with ramification index 7 over Q 2 . Let A be favorable and let B be a semistable abelian variety of conductor N d with B[2] filtered by d copies of A[2]. We obtain a class field theoretic criterion on F to guarantee that B is isogenous to A d and a fortiori, A is unique up to isogeny. [ABSTRACT FROM AUTHOR]

Based on our homological idelic class field theory, we formulate an analogue of the Hilbert reciprocity law on a rational homology 3-sphere endowed with an infinite link, in the spirit of arithmetic topology; we regard the intersection form on the unitary normal bundle of each knot as an analogue of the Hilbert symbol at each prime ideal to formulate the Hilbert reciprocity law, ensuring that cyclic covers of links are analogues of Kummer extensions. [ABSTRACT FROM AUTHOR]

For a number field F and a prime number p, the ℤp-torsion module of the Galois group of the maximal abelian pro-p extension of F unramified outside p over F, denoted by T p (F) , is an important subject in abelian p-ramification theory. In this paper, we study the group T 2 (F) = T 2 (m) of the quadratic field F = ℚ (m) . Firstly, assuming m > 0, we prove an explicit 4-rank formula for quadratic fields that rk 4 (T 2 (− m)) = rk 2 (T 2 (− m)) − rank (R) , where R is a certain explicitly described Rédei matrix over F 2 . Furthermore, using this formula, we obtain the 4-rank density formula of T 2 -groups of imaginary quadratic fields. Secondly, for l an odd prime, we obtain results about the 2-power divisibility of orders of T 2 (± l) and T 2 (± 2 l) , both of which are cyclic 2-groups. In particular, we find that # T 2 (l) ≡ 2 # T 2 (2 l) ≡ h 2 (− 2 l) (mod 16) if l ≡ 7 (mod 8), where h2(−2l) is the 2-class number of ℚ (− 2 l) . We then obtain density results for T 2 (± l) and T 2 (± 2 l) when the orders are small. Finally, based on our density results and numerical data, we propose distribution conjectures about T p (F) when F varies over real or imaginary quadratic fields for any prime p, and about T 2 (± l) and T 2 (± 2 l) when l varies, in the spirit of Cohen-Lenstra heuristics. Our conjecture in the T 2 (l) case is closely connected to Shanks-Sime-Washington's speculation on the distributions of the zeros of 2-adic L-functions and to the distributions of the fundamental units. [ABSTRACT FROM AUTHOR]

Weinberger [16] in 1972, proved that the ring of integers of a number field with unit rank at least 1 is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra [13], extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least 1 is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis holds. The aim of this paper is to show the following. Suppose that K 1 and K 2 are two Galois number fields of odd prime degree with cyclic class groups and Hilbert class fields that are abelian over Q . If K 1 K 2 is ramified over K i , then at least one K i ( i ∈ { 1 , 2 } ) must have a Euclidean ideal class. [ABSTRACT FROM AUTHOR]

We show that splitting of primes in a cubic number field with class number 3 is homogenous if and only if the degree of the splitting field of its Hilbert class field is ⩽ 2 . [ABSTRACT FROM AUTHOR]

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are 'governed' by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$. [ABSTRACT FROM AUTHOR]

Let K/k be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for K/k and the defect of weak approximation for the norm one torus \[R_{K/k}^1{\mathbb{G}_m}\]. We apply our techniques to give explicit and computable formulae for the obstruction to the Hasse norm principle and the defect of weak approximation when the normal closure of K/k has symmetric or alternating Galois group. [ABSTRACT FROM AUTHOR]

Let K be an imaginary quadratic field of discriminant dK{d}_{K} with ring of integers OK{{\mathcal{O}}}_{K}, and let τK{\tau }_{K} be an element of the complex upper half plane so that OK=[τK,1]{{\mathcal{O}}}_{K}={[}{\tau }_{K},1]. For a positive integer N, let QN(dK){{\mathcal{Q}}}_{N}({d}_{K}) be the set of primitive positive definite binary quadratic forms of discriminant dK{d}_{K} with leading coefficients relatively prime to N. Then, with any congruence subgroup Γ{\mathrm{\Gamma}} of SL2(Z){\text{SL}}_{2}({\mathbb{Z}}) one can define an equivalence relation ∼Γ{\sim }_{{\mathrm{\Gamma}}} on QN(dK){{\mathcal{Q}}}_{N}({d}_{K}). Let ℱΓ,ℚ{ {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}} denote the field of meromorphic modular functions for Γ{\mathrm{\Gamma}} with rational Fourier coefficients. We show that the set of equivalence classes QN(dK)/∼Γ{{\mathcal{Q}}}_{N}({d}_{K})/{\sim }_{{\mathrm{\Gamma}}} can be equipped with a group structure isomorphic to Gal(KℱΓ,ℚ(τK)/K)\text{Gal}(K{ {\mathcal F} }_{{\mathrm{\Gamma}},{\mathbb{Q}}}({\tau }_{K})/K) for some Γ{\mathrm{\Gamma}}, which generalizes the classical theory of form class groups.

We study the modularity of Ramanujan’s function k(τ)=r(τ)r2(2τ)k(\tau )=r(\tau ){r}^{2}(2\tau ), where r(τ)r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k(τ)k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ\tau in an imaginary quadratic field, the value k(τ)k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ1(10){{\mathrm{\Gamma}}}_{1}(10). Furthermore, we suggest a rather optimal way of evaluating the singular values of k(τ)k(\tau ) using the modular equations in the following two ways: one is that if j(τ)j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k(τ)k(\tau ), and the other is that once the value k(τ)k(\tau ) is given, we can obtain the value k(rτ)k(r\tau ) for any positive rational number r immediately.

The modular trace of the normalized Hauptmodul has been extended to the Galois trace of a class invariant by Kaneko. It is an important issue to search for class invariants for which the Galois traces have modular properties. The crucial point in this paper is that we initiate a new notion, called Siegel resolvents; we define the Siegel resolvents as the quadratic polynomials of Siegel functions of level 3, so that they are modular functions of level 3 as well. We construct real-valued class invariants over imaginary quadratic fields by using the singular values of Siegel resolvents at imaginary quadratic irrationals. We also prove that the generating series of their Galois traces become a weakly holomorphic modular form with weight 3/2. This shows that the work of D. Zagier on traces of singular moduli can be extended to the modular functions of higher level. [ABSTRACT FROM AUTHOR]

Let k be a totally real number field and p a prime. We show that the "complexity" of Greenberg's conjecture ( λ = μ = 0 ) is governed (under Leopoldt's conjecture) by the finite torsion group T k of the Galois group of the maximal abelian p-ramified pro-p-extension of k, by means of images, in T k , of ideal norms from the layers k n of the cyclotomic tower (Theorem 4.2). These images are obtained via the algorithm computing, by "unscrewing", the p-class group of k n . Conjecture 4.3 of equidistribution of these images would show that the number of steps b n of the algorithms is bounded as n → ∞ , so that (Theorem 3.3) Greenberg's conjecture, hopeless within the sole framework of Iwasawa's theory, would hold true "with probability 1". [ABSTRACT FROM AUTHOR]

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

We study a continued fraction X(τ) of order six by using the modular function theory. We first prove the modularity of X(τ), and then we obtain the modular equation of X(τ) of level n for any positive integer n; this includes the result of Vasuki et al. for n = 2, 3, 5, 7 and 11. As examples, we present the explicit modular equation of level p for all primes p less than 19. We also prove that the ray class field modulo 6 over an imaginary quadratic field K can be obtained by the value X2 (τ). Furthermore, we show that the value 1/X(τ) is an algebraic integer, and we present an explicit procedure for evaluating the values of X(τ) for infinitely many τ’s in K.

Let p≡3(mod4){p\equiv 3\pmod{4}} and l≡5(mod8){l\equiv 5\pmod{8}} be different primes such that pl=1{\frac{p}{l}=1} and 2p=pl4{\frac{2}{p}=\frac{p}{l}_{4}}. Put k=ℚ⁢(l){k=\mathbb{Q}(\sqrt{l})}, and denote by ϵ its fundamental unit. Set K=k⁢(-2⁢p⁢ϵ⁢l){K=k(\sqrt{-2p\epsilon\sqrt{l}})}, and let K2(1){K_{2}^{(1)}} be its Hilbert 2-class field, and let K2(2){K_{2}^{(2)}} be its second Hilbert 2-class field. The field K is a cyclic quartic number field, and its 2-class group is of type (2,2,2){(2,2,2)}. Our goal is to prove that the length of the 2-class field tower of K is 2, to determine the structure of the 2-group G=Gal⁡(K2(2)/K){G=\operatorname{Gal}(K_{2}^{(2)}/K)}, and thus to study the capitulation of the 2-ideal classes of K in all its unramified abelian extensions within K2(1){K_{2}^{(1)}}. Additionally, these extensions are constructed, and their abelian-type invariants are given.